Newton-Raphson method is one of the most important methods in estimating the roots of a real-valued function. As an advantage of this method we can refer to its speed of convergence. Indeed, when \(f\)  has second continuous derivative on an interval which contains the guessed root and the initial point is very close to this root, then this method converges [1]}, [2]}. To know how this method has developed through out the time we refer the reader to [3]}. This method has its own disadvantages too, among them, we can refer to its high dependence on how the initial point is chosen to start the iterate sequence. In an interval where there is a point \(a\)  with \(f^{^{\prime }}(a)=0\)  the behaviour of this sequence is quite wired and unpredictable. In this manuscripts we provide some results which make sure where the initial point should be chosen to make sure the convergence of the Newton-Raphson sequence. First, we need the following definitions and results to start our study.
